Componentwise linearity of powers of cover ideals

Let G be a finite simple graph and J(G) denote its vertex cover ideal in polynomial ring over field. The k-th symbolic power of is denoted by \(J(G)^{(k)}\). In this paper, we give criterion for ideals decomposable graphs to have the property that all their powers are not componentwise linear. Also, necessary sufficient condition on so \(J(G)^{(k)}\) linear some (equivalently, all) \(k \ge 2\) when such \(G {\setminus } N_G[A]\) has simplicial any independent set A G. Using result, prove several classes 2\). particular, if bipartite graph, then only \(J(G)^k\)